1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Prove that similar matrices have the same rank

  1. Feb 19, 2010 #1
    1. The problem statement, all variables and given/known data

    Prove that similar matrices have the same rank.


    2. Relevant equations



    3. The attempt at a solution

    Similar matrices are related via: B = P-1AP, where B, A and P are nxn matrices..
    since P is invertible, it rank(P) = n, and so since the main diagonal of P all > 0, multiplying by P will not change the rank of A, so rank B = rank A.

    Is that seem right?
     
  2. jcsd
  3. Feb 19, 2010 #2
    If you can show that rank(P-1AP) is less than or equal to rank(A), then you are done since matrix similarity is symmetric (if A is similar to B, then B is similar to A).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook